Otto Cycle Efficiency and Compression Ratio

The efficiency of an Otto cycle is a key parameter that indicates how effectively the cycle converts heat energy into useful work. This efficiency is directly related to the compression ratio and the ratio of specific heats of the working fluid.

Otto Cycle Efficiency Formula

The thermal efficiency ($\eta$) of an ideal Otto cycle is given by the following formula:

$$\eta = 1 - \frac{1}{r^{\gamma-1}}$$

Where:

• $\eta$ is the thermal efficiency of the Otto cycle.
• $r$ is the compression ratio, which is the ratio of the maximum volume to the minimum volume during the compression stroke.
• $\gamma$ (gamma) is the ratio of specific heats ($C_p / C_v$) for the working fluid.

Given Values in the Problem

From the question, we are provided with the following information:

• Efficiency of the Otto cycle ($\eta$) = 50% = 0.50
• Ratio of specific heats ($\gamma$) = 1.25

Calculating the Compression Ratio

To find the value of the compression ratio ($r$), we need to substitute the given values into the Otto cycle efficiency formula and solve for $r$.

Let's substitute the values:

$$0.50 = 1 - \frac{1}{r^{1.25-1}}$$

First, simplify the exponent:

$$0.50 = 1 - \frac{1}{r^{0.25}}$$

Now, rearrange the equation to isolate the term involving $r$:

$$\frac{1}{r^{0.25}} = 1 - 0.50$$

$$\frac{1}{r^{0.25}} = 0.50$$

Next, take the reciprocal of both sides to get $r^{0.25}$:

$$r^{0.25} = \frac{1}{0.50}$$

$$r^{0.25} = 2$$

We know that $0.25$ is equivalent to $\frac{1}{4}$. So, the equation becomes:

$$r^{\frac{1}{4}} = 2$$

To find $r$, we need to raise both sides of the equation to the power of 4:

$$(r^{\frac{1}{4}})^4 = 2^4$$

$$r = 2 \times 2 \times 2 \times 2$$

$$r = 16$$

Therefore, the value of the compression ratio is 16.

Summary of Calculation

This result shows how the efficiency of an Otto cycle is determined by its compression ratio and the properties of the working fluid.